K₁ and the Whitehead Group

The group $K_1(R)$ captures information about automorphisms of free modules, specifically the "non-elementary" part of the general linear group. It was introduced by Bass and serves as the algebraic analogue of the Whitehead torsion invariant from topology.

The general linear group and elementary matrices

Definition1.4Stable general linear group

For a ring $R$ , the general linear group $GL_n(R)$ is the group of invertible $n \times n$ matrices over $R$ . The inclusions $GL_n(R) \hookrightarrow GL_{n+1}(R)$ via $A \mapsto \begin{pmatrix} A & 0 \\ 0 & 1 \end{pmatrix}$ yield the stable general linear group

$GL(R) = \varinjlim_{n} GL_n(R) = \bigcup_{n=1}^{\infty} GL_n(R).$

Definition1.5Elementary matrices and E(R)

An elementary matrix $e_{ij}(\lambda)$ for $i \neq j$ and $\lambda \in R$ is the matrix $I + \lambda E_{ij}$ , where $E_{ij}$ is the matrix with $1$ in position $(i,j)$ and $0$ elsewhere. The subgroup generated by all elementary matrices in $GL_n(R)$ is denoted $E_n(R)$ , and

$E(R) = \varinjlim_{n} E_n(R) = \bigcup_{n=1}^{\infty} E_n(R) \subseteq GL(R).$

Elementary matrices satisfy the Steinberg relations:

$e_{ij}(\lambda) \cdot e_{ij}(\mu) = e_{ij}(\lambda + \mu)$
$[e_{ij}(\lambda), e_{kl}(\mu)] = e_{il}(\lambda\mu)$ when $j = k$ and $i \neq l$
$[e_{ij}(\lambda), e_{kl}(\mu)] = 1$ when $j \neq k$ and $i \neq l$

Definition of K₁

Definition1.6K₁ of a ring

The first algebraic K-group of $R$ is defined as

$K_1(R) = GL(R) / E(R) = GL(R)^{\mathrm{ab}} / \text{(image of } E(R)\text{)}.$

By the Whitehead lemma, $E(R) = [GL(R), GL(R)]$ is the commutator subgroup of $GL(R)$ . Thus

$K_1(R) = GL(R) / [GL(R), GL(R)] = H_1(GL(R); \mathbb{Z}),$

the abelianization of $GL(R)$ .

ExampleK₁ of a commutative ring

For a commutative ring $R$ , the determinant map $\det: GL_n(R) \to R^{\times}$ is compatible with stabilization and sends elementary matrices to $1$ , inducing a surjection

$\det: K_1(R) \twoheadrightarrow R^{\times}.$

The kernel $SK_1(R) = \ker(\det)$ is called the special K₁ group. Thus $K_1(R) \cong R^{\times} \oplus SK_1(R)$ .

For fields, PIDs, and local rings, $SK_1(R) = 0$ , so $K_1(R) \cong R^{\times}$ . For example:

$K_1(\mathbb{Z}) \cong \mathbb{Z}^{\times} = \{+1, -1\} \cong \mathbb{Z}/2\mathbb{Z}$
$K_1(F) \cong F^{\times}$ for any field $F$
$K_1(k[x_1, \ldots, x_n]) \cong k^{\times}$ for a field $k$

ExampleK₁ of number rings

For the ring of integers $\mathcal{O}_F$ of a number field $F$ , the Dirichlet unit theorem implies

$K_1(\mathcal{O}_F) \cong \mathcal{O}_F^{\times} \cong \mu(F) \times \mathbb{Z}^{r_1 + r_2 - 1}$

where $\mu(F)$ is the group of roots of unity in $F$ , $r_1$ is the number of real embeddings, and $r_2$ is the number of pairs of complex embeddings. Here $SK_1(\mathcal{O}_F) = 0$ by Bass--Milnor--Serre for number fields.

The Whitehead group

Definition1.7Whitehead group

For a group $G$ and a ring $R$ , the Whitehead group is

$\operatorname{Wh}(G) = K_1(\mathbb{Z}[G]) / \{\pm g : g \in G\}.$

This quotients out the "obvious" units $\pm g \in \mathbb{Z}[G]^{\times}$ . The Whitehead group is a topological invariant: for a connected CW complex $X$ with $\pi_1(X) = G$ , the Whitehead torsion of an $h$ -cobordism lives in $\operatorname{Wh}(G)$ .

ExampleWhitehead groups of finite abelian groups

For $G = \mathbb{Z}/n\mathbb{Z}$ :

$\operatorname{Wh}(\mathbb{Z}/1) = 0$ (trivially)
$\operatorname{Wh}(\mathbb{Z}/2) = 0$ , $\operatorname{Wh}(\mathbb{Z}/3) = 0$ , $\operatorname{Wh}(\mathbb{Z}/4) = 0$
$\operatorname{Wh}(\mathbb{Z}/5) \cong \mathbb{Z}$ (generated by the golden ratio unit $(1+\sqrt{5})/2$ in $\mathbb{Z}[\zeta_5]$ )

For free abelian groups, $\operatorname{Wh}(\mathbb{Z}^n) = 0$ for all $n$ (a deep result). The Bass--Heller--Swan theorem gives a splitting $K_1(R[t, t^{-1}]) \cong K_1(R) \oplus K_0(R) \oplus NK_1(R) \oplus NK_1(R)$ where $NK_1$ are nil-groups.

RemarkConnection to topology

The $s$ -cobordism theorem (Smale, Barden, Mazur, Stallings) states that an $h$ -cobordism $(W; M_0, M_1)$ of dimension $\geq 6$ is trivial (i.e., $W \cong M_0 \times [0,1]$ ) if and only if its Whitehead torsion $\tau(W, M_0) \in \operatorname{Wh}(\pi_1(M_0))$ vanishes. When $\operatorname{Wh}(\pi_1) = 0$ (e.g., for simply connected manifolds), every $h$ -cobordism is trivial --- this proves the Poincare conjecture in dimensions $\geq 5$ .